mathBook

1. 1 1. 1 2. 4 3. 8 4. 11 5. 15

2. 19 1. 19 2. 21

23

25

iii

1

1.

Nakhl H. Asmar- Partial Differential Equations with Fourier Seriesand Boundary Value problems ( , Pearson Prentice Hall 2005,2000) . .

. .

, ;

f(x), f(x, y, z), u(x, t)

. .

;

limn fn(x),

n=1

un(x, t)

. /pointwise /uniform

. L2-

. ;

,

ex, sinx, cosx

A.4, A.5 .

( ) ,

1

2 1

u(x) + p(x)u(x) = g(x)

; (, )

u(x) + pu(x) + qu(x) = 0 .

A.1, A.2 .

. 2-

a0 +n=1

(an cos(nx) + bn sin(nx))

. , . , .

( ), , .

: u(x, t) . ux(x, t) = t- x- ut (x, t) = x- t- .

1.1.

u(x, t) := x2 + sin(t)x

u

x(x, t) = 2x+ sin(t)

u

t(x, t) = cos(t)x.

.

1.2. (1) f = f(x), g = g(s)

f(s) := f(g(s))

df

ds(s) =

df

dx(g(s)) dg

ds(s).

1 3

(2) f = f(x), g = g(s, t)

f(s) := f(g(s, t))

f

s(s) =

df

dx(g(s)) g

s(s).

(3)

u = u(,), = (x, t), = (x, t)

u(x, t) := u((x, t),(x, t))

u

x=

u

x

+u

x

u

t=

u

t

+u

t

.

- : u = u(x, t)u

x+

u

t= 0. (1.1)

?(1) :

u(x, t) := x t.

u

x= 1,

u

t= 1

u

x+

u

t= 0.

(2) , f -

u(x, t) := f(x t) (1.2) :

u

x= f ,

u

t= f .

(t s)3, e(ts)2 , sin(t s) + 3

.

4 1

: , C- y(t) := C eat y = ay C = y(0).

u(x, 0) = (1.3)

/initial value condition .- /initialvalue problem .

(1.1) , f u = f(x t) .

2.

(1.1) .

u(x, t) = v(,), (x, t) = ax+ bt, (x, t) = cx+ dt

( adbc = 0 ).

u

x= a

v

+ c

v

u

t= b

v

+ d

v

u

x+

u

t= (a+ b)

v

+ (c+ d)

v

.

a = 1, b = 0, c = 1, d = 1

v

(,) = 0

. , - C =C()-

v(,) = C() . (x, t) = x t

u(x, t) = v(,) = C() = C(x t) (1.2) .

1.3. - - . .

1.4. (1.1) /transport equation . advection equation .

u

x+ (x, t)

u

t= k(x, t)

2 5

. 1.1- The method of characteristic curves 1.2.27- .

. .

:

1.5. c > 0 . : u = u(x, t)

2u

t2 c2

2u

x2= 0

: u = u(x, t)

u

t c2

2u

x2= 0

: u = u(x, y)

2u

x2+ c2

2u

y2= 0.

, ,

2 c22 = a c22 = a2 + c22 = a

, , . .

:

2u

t2 c2

2u

x2= 0. (1.4)

x [0, L] , t R- u(0, t) = 0 = u(L, t) (1.5)

. L . . /boundary condition .

: :

u

t cu

x= 0

6 1

. , F ,G 2

u(x, t) = F (x+ ct) +G(x ct) . .

: . . .

. . .

c = 1, L =

. 2u

t2

2u

x2= 0

u(0, t) = 0 = u(, t)

. 1:

u(x, t) := sin(x) cos(t).

2u

t2= u =

2u

x2

2u

t2

2u

x2u = 0.

ut(x, t) :=

u

t(x, t) = sin(x) sin(t)

Iu(x, 0) = sin(x)ut(x, 0) = 0

. Iu(x, 0) = 0ut(x, 0) = sin(x)

u(x, t) := sin(x) sin(t)

1 .

2 7

. , ,Iu(x, 0) = 12 sin(x)ut(x, 0) = 3 sin(x)

u(x, t) :=12sin(x) cos(t) + 3 sin(x) sin(t)

. ( .) , I

u(x, 0) = f(x)ut(x, 0) = g(x)

Iu(x, 0) = f(x)ut(x, 0) = 0

Iu(x, 0) = 0ut(x, 0) = g(x)

. .

c, L- , :

u(x, t) := sin3

Lx4cos

3c

Lt4.

/principal solution . : c = 1, L =

u2(x, t) := sin(2x) cos(2t)...

un(x, t) := sin(nx) cos(nt) .

2unt2

= n2u = 2u

x2

. u(0, t) = 0 = u(, t) . I

un(x, 0) = sin(nx)unt (x, 0) = 0

. c, L-

un(x, t) := sin3n

Lx4cos

3cn

Lt4

. /normal mode .,

u(x, t) :=Nn=1

bn sin3n

Lx4cos

3cn

Lt4

8 1

. /superposition . :

u(x, 0) =Nn=1

bn sin3n

Lx4

ut(x, 0) = 0

.

ut(x, t) =Nn=1

3cn

L

4bn sin

3n

Lx4sin3cn

Lt4

.

3.

u(x, 0) = f(x)

f(x) =Nn=1

bn sin3n

Lx4

u(x, t) =Nn=1

bn sin3n

Lx4cos

3cn

Lt4

.

f(x) =n=1

bn sin3n

Lx4

u(x, t) =n=1

bn sin3n

Lx4cos

3cn

Lt4

, ? ? .

, . o . . 3.5 . ,

u

t c2

2u

x2= 0

u(0, t) = 0 = u(L, t)u(x, 0) = f(x)

. L , 0 , x f(x) .

3 9

u1(x, t) := sin3

Lx4ec

2( L)2t

...

un(x, t) := sin3n

Lx4ec

2(nL )2t

u(x, t) :=n=1

bn sin3n

Lx4ec

2(nL )2t

u(x, 0) :=n=1

bn sin3n

Lx4

. , f(x)-

f(x) =n=1

bn sin3n

Lx4

(1.6)

. f - , , ? . (1.6) , . , .

2.1 .

1.6. f : R R x R-

f(x+ T ) = f(x) f - T - .

T - [0, T ) ( a R- [a, a+ T ) (a, a+ T ]) .

1.7. f(x) = x (L,L] 2L :f(x) = x 2kL, x ((2k 1)L, (2k + 1)L], k Z.

. . . ,L,L, 3L, . . . . .

1.8. 18- /saw-tooth : 2- [0, 2)

f(x) :=12( x), x [0, 2)

10 1

o.

.

1.9. f : [a, b] R x- f(x) := lim

0,>0 f(x ), f(x+) := lim0,>0 f(x+ )

x- . f(x) f - x . x a b f(a+) f(b)- .

1.10. f : [a, b] R f - /piecewise contin-uous .

, [a, b]

a = t0 < t1 < < tm < tm+1 = b f (ti, ti+1) f(ti), f(ti+) f - ( a, b f(a+), f(b)- ). .

C0 = C1 = , ,

. C1 /smooth C .

.

1.11. f : [a, b] R f f 2 f - C1 . f : R R C1 f - C1 .

1.12. (1.7, 1.8) C1 .

1.13. f(x) = x2 [1, 1] 2- . f C1.

2 f f - .

4 11

4.

, 3, . .

. .

1.14. f : R R T - , a R- T

0f(x)dx =

a+Ta

f(x)

. . f =

f(x) T g(x) := f(px), p = 0 T/p . sin(x), cos(x) 2, sin(nx), cos(nx) 2/n . n , sin(nx), cos(nx) 2 .

1.15. n,m . cos(nx)dx =

I2, n = 0,0, n = 0

sin(nx)dx = 0, n

. ,

cos(nx) cos(mx)dx =

2, n = m = 0,, n = m = 0,0, n = m

sin(nx) cos(mx)dx = 0, n, m

sin(nx) sin(mx)dx =

I, n = m,0, n = m

.

. :

1dx = 2,

3 .

12 1 cos(nx)dx =

1n(sin(n) sin(n)) = 0, n = 0,

sin(nx)dx =

0(sin(nx) + sin(nx))dx = 0.

:

cos(x) cos(y) =12(cos(x y) + cos(x+ y))

sin(x) cos(y) =12(sin(x y) + sin(x+ y))

sin(x) sin(y) =12(cos(x y) cos(x+ y)).

1.16. m = 0 .

1.17. [0, 2] .

2 f : R R

f(x) = a0 +n=1


f(x)dx =

a0dx+n=1

(an cos(nx)dx+ bn

sin(nx)dx)

= 2a0

. ,m = 0 ,

f(x) cos(mx)dx =

a0 cos(mx)dx+n=1

(an cos(nx) cos(mx)dx

+ bn sin(nx) cos(mx)dx)

= am,

f(x) sin(mx)dx =

a0 sin(mx)dx+n=1

(an cos(nx) sin(mx)dx

+ bn sin(nx) sin(mx)dx)

= bm

( ). .

4 13

1.18. f : R R 2-

a0 :=12

f(x)dx

an :=1

f(x) cos(nx)dx

bn :=1

f(x) sin(nx)dx

.

a0 +n=1


- f -

sN = sN (f) := a0 +Nn=1


.

. , , .

1.19. f [,] e . ,

f(x) cos(nx), f(x) sin(nx)

.4 .

1.20. [0, 2] .

1.21. (1.8): 2-

f(x) :=12( x), x [0, 2).

, f [,] ( 1.29- )

a0 =12

f(x)dx = 0

an =1

f(x) cos(nx)dx = 0

4 - f f2 , f L2[,], .

14 1

bn =1

20

12( x) sin(nx)dx

=12

20

(x) sin(nx)dx

=12

20

1nxd cos(nx)

=12

3 1nx cos(nx)

----20 20

1ncos(nx)dx

4=

1n

. f - n=1

1nsin(nx)

. ? , , Ix (0, 2) f(x)-x = 0 0-

.

. 2.8 .5

T 1.22. f : R R 2- C1 . x R-

a0 +n=1

(an cos(nx) + bn sin(nx)) =12(f(x) + f(x+))

. f x f(x) = f(x) = f(x+)

a0 +n=1

(an cos(nx) + bn sin(nx)) = f(x)

.

1.23. f C1 f - .

5 Asmar- .

5 15

5.

2- C1 f : R R 1.22- f - f - . .

T 1.24. f : R R 2- C1 . f - f - .

2.9 .

1.25. C1 . , . .

http://www.sosmath.com/fourier/fourier3/gibbs.html

f(x) :=I1, x [, 0)1, x [0,)

4

3sin(x) +

13sin(3x) +

15sin(5x) + . . .

4 0

s2n13

2n

4 2

0

sin(x)x

dx 1.18, n

.

1.26. :

g(x) :=I + x, x [, 0] x, x [0,]

C1 .

2+

4

3cos(x) +

132cos(3x) +

152cos(5x) + . . .

4 . x- ----cos(2n 1)x(2n 1)2

---- 1(2n 1)2 , n 1

n=1

1(2n 1)2

16 1

1.27. 2- . ,x = 0

=

2+

4

n=1

1(2n 1)2

2

8=

n=1

1(2n 1)2

. ,n=1

1n2

=n=1

1(2n 1)2 +

n=1

1(2n)2

=n=1

1(2n 1)2 +

14

n=1

1n2

2

6=

n=1

1n2

.

1.28. (1.8, 1.26)

f(x) +12g(x) =

I x, x [0,)0, x [, 2)

.

4+

n=1

31 (1)nn2

cos(nx) +1nsin(nx)

4 .

. .

1.29. f : R R x- f(x) = f(x) , x- f(x) = f(x) .

1.30. |x|, x2, cos(nx) , x, x3, sin(nx) .f p

pf(x)dx =

p0f(x) + f(x)dx = 0

. 1.21- .

.

5 17

1.31. f

f :=12(f(x) + f(x+))

.

, f f - . 1.22-oo f - f - .

1.32. f : R R 2 C1 f -

f(x) = a0 +n=1


. (i) f n- bn = 0,(ii) f n- an = 0.

. f - f - .

1.33.

a0 +n=1

an cos(nx)

,n=1

bn sin(nx)

.

1.34. A.2(1, 5, 9, 62), 1.1(2a, 3a, 4), 1.2(1, 2, 3, 4, 16, 18, 25), 2.1(3,4, 5, 27), 2.2(2, 4, 6a, 7a, 20a).6

6 . .

2

1.

1.32- .

2.1. ( 1.8)

n=1

sin(nx)n

.

2.2. ( 1.26)

2+

4

n=1

cos((2n 1)x)(2n 1)2 .

[0,] [,]- .

2.3. f : [0,] R- (,]-.f(x) := f(|x|)

f(x) :=

f(x), x (, 0)f(x), x (0,)0, x = 0,

2- R- . f , f . f(x) =

x, x [0,] , .

f

a0 =12

f(x)dx

=1

0f(x)dx

an =1

f(x) cos(nx)dx

=2

0f(x) cos(nx)dx

19

20 2

bn =1

f(x) sin(nx)dx

=2

0f(x) sin(nx)dx

.

a0 +n=1

an cos(nx)

f - ,n=1

bn sin(nx)

f - .

2.4. f C1 f C1 f - f - . f(0) = f() = 0 , f .

O 2- . f 2p, p > 0 .

g(x) := f3p

x4

g 2 :

g(x+ 2) = f3p

(x+ 2)

4= f

3p

x+ 2p

4= f

3p

x4= g(x).

f(x) = g1px2 g- f -

. f C1 g

f(x) = g3

px4= a0 +

n=1

3an cos

3

pnx4+ bn sin

3

pnx44

.

a0 =12

g(x)dx

=12

f3p

x4dx

=12p

pp

f(x)dx

an =1p

pp

f(x) cos3

pnx4dx

bn =1p

pp

f(x) sin3

pnx4dx

2 21

. , . 2 2p .

1, cos3

px4, cos

3

p2x4, . . . , sin

3

px4, . . .

ppcos

3

px42

dx = p

.

2.5. p = 2 . f : [0,] R

a0 +n=1

(an cos(2nx) + bn sin(2nx))

: - .

f(x) = sin(x) [0,] . sin(x) sin(x) . sin(x)- -

n=0

an cos(2nx)

. , f(x) = x cos(x), x [/2,/2]

n=1

bn sin(2nx)

.

2.6. h(x) = 1x2, x [1, 1]2- . , C1

h(x) =23 4

2

n=1

(1)nn2

cos(nx)

.

2.

2.5 .

a0 +n=1


22 2

N

sN := a0 +Nn=1


. . L2- .

2.7. {fn}n=1 f [a, b] . N b

a|fN (x) f(x)|2dx 0

fn- f - L2- .1

2.8. f .

EN :=12

|f(x) sN (x)|2dx . EN - f - N .

.

T 2.9. f : R R 2 , . N EN 0, f - f -L2-.

2.10. f C1 f - L2-. . f - .2

.

T 2.11. f .

EN =12

f(x)2dx a20 12

n=1

(a2n + b2n)

.

1 fn f L2- .2 f L2[,]

.

[Asm05] Nakhl H. Asmar, Partial differential equations with fourier series and boundary valueproblems, 2 ed., Pearson Prentice Hall, 2005.

23

C0, 10C1, 10C, 10

equationadvection, 4transport, 4

normal mode, 7

principal solution, 7

superposition, 8

, 15, 2, 1 , 4 , 4 , 7 , 7 , 5 , 4 , 3 , 17, 20

, 17, 20, 8, 9 , 2 , 2, 5, 5, 5, 8, 4

, 15, 11, 16 , 10, 16

, 10 C1, 10 , 9

, 10

, 9

25

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